More generally, the radical of an ideal in can be defined over an arbitrary ring. Let I be an ideal of a ring R, the radical of I is the set of a∈R such that every m-system containing a has a non-empty intersection with I:

I:={a∈R∣if ⁢S⁢ is an m-system, and ⁢a∈S, then ⁢S∩I≠∅}.

Under this definition, we see that I is again an ideal (two-sided) and it is a subset of {a∈R⁢∣an∈I⁢ for some integer ⁢n>⁢0}. Furthermore, if R is commutative, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalization” of the radical of an ideal in a commutative ring.